Artin groups of types F4 and H4 are not commensurable with that of type D4

Property $R_\infty$ for some spherical and affine Artin-Tits groups

“…It was shown in [Sor20a, Th. 1] that Γ is isomorphic to the pure orientation preserving mapping class group of Σ Γ and in [Sor20b,Cor. 6] that Aut(Γ) is isomorphic to the extended mapping class group Mod ± (Σ Γ ), which contains Γ as a subgroup of index 12.…”

“…Hyperbolicity of C parab (A(X)) was established for X being A n (n 3), B n (n 3), A n (n 2), and C n (n 2), see [CC21]. Also, it can be shown using results from [Sor20b] that C parab (A(D 4 )) is isomorphic to the curve graph of the three times punctured torus, and hence it is also hyperbolic by Masur-Minsky's theorem. Given a non-inner automorphism ϕ of A(X), we found that in some cases, the action of A(X) extends to an action of A(X) ϕ on C parab (A(X)); this is for instance the case when ϕ is a parabolic-preserving automorphism.…”